\documentclass{article}
\usepackage{ctex}
    
\author{uncle-lu}
\title{NOIP2016Day2愤怒的小鸟数学推导}
\date{\today}

\begin{document}

\maketitle

\newpage

    已知$(x_1,y_1) (x_2,y_2)$与\(y=ax^2 + bx\)

    将$(x_1,y_1) (x_2,y_2)$代入方程中可得
    \begin{displaymath}
        \left\{ \begin{array}{ll}
            y_1= a x_1^2 + b x_1 & \textcircled{1}\\
            y_2= a x_1^2 + b x_2 & \textcircled{2} 
        \end{array} \right.
    \end{displaymath}
    由\textcircled{1}可得
    \[
    y_1 = a x_1^2 + b x_1 \\
    \]
    \[
        b = 
        \frac{ y_1-a x_1^2 }{ x_1 } 
        \qquad \textcircled{3}
    \]

    将\textcircled{3}代入\textcircled{2}中

    \[
    y_2=a x_2^2 +
    \frac{ y_1-a x_1^2 }{ x_1 }
    x_2 
    \]
    \[
    y_2=a x_2^2 +(
    \frac{ y_1 }{ x_1 }
    - a
    \frac{ x_1^1 }{ x_1 }
    ) x_2 
    \]
    \[
    y_2=a x_2^2 +
    \frac{ x_2 y_1 }{ x_1 } 
    - a x_1 x_2 
    \]
    \[
    y_2 - 
    \frac{ x_2 y_1 }{ x_1 } 
    = ( x_2^2 - x_1 x_2) a 
    \]
    \[
    a = 
    \frac{ y_2- 
    \frac{ x_2 y_1 }{ x_1 }
    }{ x_2^2 - x_1 x_2 } 
    \]

    于是乎我们很轻松就可以算出a与b.

\end{document}